Energy available in fuel

I've been confusing myself over how much energy it takes to accelerate a vehicle (assuming no friction/air) with a given amount of fuel.

I know that the amount of energy required to reach a given speed increases quadratically, but does the amount of fuel required also increase quadratically? Or does the amount of energy taken from the fuel increase over time?

I'm pretty sure this is me confusing impulse with energy, but I don't know the relevant details of how a car works to begin with. For example, you press the gas the same amount and for the same amount of time to go from 10 to 20 as you would to go from 50 to 60 (so same impulse). Does it use the same amount of fuel?

For example, would you press the gas the same amount and for the same amount of time to go from 10 to 20 as you would to go from 50 to 60 (so same impulse).

No, it would take more pedal pressure or more time to accelerate from 50 to 60 than it would from 10 to 20. Ignoring aerodynamic drag factor, power = force times speed, for example, horsepower = force (lbs) times speed (mph) divided by 375 (conversion factor). So even if the force is the same, the speed at 50 is five times as much as it is at 10, so it takes 5 times the power to generate the same force at 50 as it does at 10.

On the other hand, a typical gas engine is most efficient at a certain load and rpm. A typical medium or larger sized car gets it's best milage at around 45mph, a combination of gearing, rpm, and engine load.